Worksheet: Linear Transformation Composition

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In this worksheet, we will practice finding the matrix of two or more consecutive linear transformations.

Q1:

Find the matrix with respect to the standard basis vectors for the linear transformation that rotates every vector in ℝ through an angle of 𝜋3 and then reflects it across the 𝑦-axis.

  • A⎡⎢⎢⎢⎣−12√32√3212⎤⎥⎥⎥⎦
  • B⎡⎢⎢⎢⎣12√32√3212⎤⎥⎥⎥⎦
  • C⎡⎢⎢⎢⎣12−√32√3212⎤⎥⎥⎥⎦
  • D⎡⎢⎢⎢⎣−√321212√32⎤⎥⎥⎥⎦
  • E⎡⎢⎢⎢⎣−√3212√3212⎤⎥⎥⎥⎦

Q2:

A linear transformation is formed by reflecting every vector in ℝ in the 𝑦-axis and then rotating the resulting vector through an angle of 𝜋4. Find the matrix of this linear transformation.

  • A⎡⎢⎢⎢⎣−√22√22−√22√22⎤⎥⎥⎥⎦
  • B⎡⎢⎢⎢⎣−√2200√22⎤⎥⎥⎥⎦
  • C⎡⎢⎢⎢⎣−√22√22√22√22⎤⎥⎥⎥⎦
  • D⎡⎢⎢⎢⎣−√22−√22−√22√22⎤⎥⎥⎥⎦
  • E⎡⎢⎢⎢⎣√22−√22√22√22⎤⎥⎥⎥⎦

Q3:

A linear transformation is formed by rotating every vector in ℝ through an angle of 𝜋6, reflecting the resulting vector in the 𝑥-axis, and finally reflecting this vector in the 𝑦-axis. Find the matrix of this linear transformation.

  • A⎡⎢⎢⎢⎣−√3200√32⎤⎥⎥⎥⎦
  • B⎡⎢⎢⎢⎣−√32−1212√32⎤⎥⎥⎥⎦
  • C⎡⎢⎢⎢⎣√32−1212√32⎤⎥⎥⎥⎦
  • D⎡⎢⎢⎢⎣−√3212−12−√32⎤⎥⎥⎥⎦
  • E⎡⎢⎢⎢⎣√32−12−12−√32⎤⎥⎥⎥⎦

Q4:

A linear transformation is formed by rotating every vector in ℝ through an angle of 2𝜋3 and then reflecting the resulting vector in the 𝑥-axis. Find the matrix of this linear transformation.

  • A⎡⎢⎢⎢⎣−12−√32√32−12⎤⎥⎥⎥⎦
  • B⎡⎢⎢⎣−120012⎤⎥⎥⎦
  • C⎡⎢⎢⎢⎣12−√32−√32−12⎤⎥⎥⎥⎦
  • D⎡⎢⎢⎢⎣−12√32−√3212⎤⎥⎥⎥⎦
  • E⎡⎢⎢⎢⎣−12−√32−√3212⎤⎥⎥⎥⎦

Q5:

A linear transformation is formed by reflecting every vector in ℝ across the 𝑦-axis and then rotating the resulting vector through an angle of 𝜋6. Find the matrix of this linear transformation.

  • A⎡⎢⎢⎢⎣−√3200√32⎤⎥⎥⎥⎦
  • B⎡⎢⎢⎢⎣−√32−12−12√32⎤⎥⎥⎥⎦
  • C⎡⎢⎢⎢⎣√32−1212√32⎤⎥⎥⎥⎦
  • D⎡⎢⎢⎢⎣−√321212√32⎤⎥⎥⎥⎦
  • E⎡⎢⎢⎢⎣−√3212−12√32⎤⎥⎥⎥⎦

Q6:

A linear transformation is formed by rotating every vector in ℝ through an angle of 𝜋3 and then reflecting the resulting vector in the 𝑥-axis. Find the matrix of this linear transformation.

  • A⎡⎢⎢⎣1200−12⎤⎥⎥⎦
  • B⎡⎢⎢⎢⎣12−√32√3212⎤⎥⎥⎥⎦
  • C⎡⎢⎢⎢⎣−12−√32−√3212⎤⎥⎥⎥⎦
  • D⎡⎢⎢⎢⎣12−√32−√32−12⎤⎥⎥⎥⎦
  • E⎡⎢⎢⎢⎣12√32−√32−12⎤⎥⎥⎥⎦

Q7:

A linear transformation is formed by reflecting every vector in ℝ in the 𝑥-axis and then rotating the resulting vector through an angle of 𝜋6. Find the matrix of this linear transformation.

  • A⎡⎢⎢⎢⎣√3200−√32⎤⎥⎥⎥⎦
  • B⎡⎢⎢⎢⎣√321212−√32⎤⎥⎥⎥⎦
  • C⎡⎢⎢⎢⎣√32−1212−√32⎤⎥⎥⎥⎦
  • D⎡⎢⎢⎢⎣√32−12−12−√32⎤⎥⎥⎥⎦
  • E⎡⎢⎢⎢⎣√32−1212√32⎤⎥⎥⎥⎦

Q8:

A linear transformation is formed by reflecting every vector in ℝ in the 𝑥-axis and then rotating the resulting vector through an angle of 𝜋4. Find the matrix of this linear transformation.

  • A⎡⎢⎢⎢⎣√22−√22−√22−√22⎤⎥⎥⎥⎦
  • B⎡⎢⎢⎢⎣√2200−√22⎤⎥⎥⎥⎦
  • C⎡⎢⎢⎢⎣√22√22√22−√22⎤⎥⎥⎥⎦
  • D⎡⎢⎢⎢⎣√22−√22√22−√22⎤⎥⎥⎥⎦
  • E⎡⎢⎢⎢⎣√22−√22√22√22⎤⎥⎥⎥⎦

Q9:

Let the matrix 𝐴 represent rotation in the plane through an angle of 𝜃 and let the matrix 𝐵 represent reflection in the 𝑥-axis.

What is the matrix 𝐴?

  • A𝜃𝜃−𝜃𝜃cossinsincos
  • B𝜃𝜃𝜃𝜃cossinsincos
  • C𝜃−𝜃𝜃𝜃sincoscossin
  • D𝜃−𝜃𝜃𝜃cossinsincos
  • E𝜃𝜃𝜃𝜃sincoscossin

What is the matrix 𝐵?

  • A−100−1
  • B−1001
  • C0110
  • D0−1−10
  • E100−1

What is the matrix 𝐴𝐵?

  • A𝜃𝜃𝜃−𝜃cossinsincos
  • B𝜃−𝜃𝜃𝜃cossinsincos
  • C𝜃−𝜃𝜃𝜃sincoscossin
  • D𝜃𝜃−𝜃𝜃cossinsincos
  • E𝜃𝜃𝜃−𝜃sincoscossin

Q10:

Suppose that the matrix 𝐴 represents rotation about the origin through an angle of 𝜃 (measuring between 0∘ and 90∘) and the matrix 𝐵 represents reflection in the 𝑥-axis.

Find 𝑀=𝐴𝐵.

  • A𝑀=𝐴𝐵=−𝜃𝜃−𝜃𝜃cossinsincos
  • B𝑀=𝐴𝐵=−𝜃−𝜃𝜃−𝜃cossinsincos
  • C𝑀=𝐴𝐵=𝜃𝜃𝜃−𝜃cossinsincos
  • D𝑀=𝐴𝐵=𝜃−𝜃𝜃−𝜃cossinsincos
  • E𝑀=𝐴𝐵=𝜃−𝜃𝜃𝜃cossinsincos

Note that 𝑀 is a reflection in a line through the origin. Let this line of reflection have equation 𝑦=𝑘𝑥. By considering the image of the vector ⟨1,0⟩, determine the measure of the angle between the 𝑥-axis and the line of reflection.

  • A𝜋2−𝜃
  • B𝜃2
  • C−𝜃2
  • D𝜃
  • E−𝜃

What, therefore, is the slope 𝑘 of the line of reflection?

  • Atan𝜃
  • Btan𝜃2
  • C−𝜃tan
  • D−𝜃2tan
  • Ecot𝜃

If v lies in the direction of the line of reflection, what is 𝑀v?

  • A⟨1,0⟩
  • B−v
  • C⟨0,1⟩
  • Dv
  • E⟨0,0⟩

By solving the equation obtained in the previous part, find another expression for the slope 𝑘 of the line of reflection.

  • A𝑘=𝜃1+𝜃=1−𝜃𝜃sincoscossin
  • B𝑘=𝜃1−𝜃=1+𝜃𝜃cossinsincos
  • C𝑘=𝜃1−𝜃=1+𝜃𝜃sincoscossin
  • D𝑘=𝜃1−𝜃=1−𝜃𝜃sincoscossin
  • E𝑘=𝜃1+𝜃=1−𝜃𝜃cossincossin

Q11:

A linear transformation is formed by rotating every vector in ℝ through an angle of 30∘ counterclockwise (when viewed from the positive 𝑧-axis) about the 𝑧-axis and then reflecting the resulting vector in the 𝑥𝑦-plane. Find the matrix of this linear transformation.

  • A⎡⎢⎢⎢⎢⎢⎣√32−12012√32000−1⎤⎥⎥⎥⎥⎥⎦
  • B⎡⎢⎢⎢⎢⎢⎣√32−12012√320001⎤⎥⎥⎥⎥⎥⎦
  • C⎡⎢⎢⎢⎢⎢⎣12−√320√32120001⎤⎥⎥⎥⎥⎥⎦
  • D⎡⎢⎢⎢⎢⎢⎣12−√320√3212000−1⎤⎥⎥⎥⎥⎥⎦
  • E⎡⎢⎢⎢⎢⎢⎣√32−120−12−√32000−1⎤⎥⎥⎥⎥⎥⎦

Q12:

A linear transformation is formed by rotating every vector in ℝ through an angle of 𝜋4 and then reflecting the resulting vector in the 𝑥-axis. Find the matrix of this linear transformation.

  • A⎡⎢⎢⎢⎣√2200−√22⎤⎥⎥⎥⎦
  • B⎡⎢⎢⎢⎣−√22√22√22√22⎤⎥⎥⎥⎦
  • C⎡⎢⎢⎢⎣√22√22√22√22⎤⎥⎥⎥⎦
  • D⎡⎢⎢⎢⎣√22−√22√22√22⎤⎥⎥⎥⎦
  • E⎡⎢⎢⎢⎣√22−√22−√22−√22⎤⎥⎥⎥⎦

Q13:

A vector in ℝ is rotated counterclockwise about the origin through an angle of 2𝜋3, and the result is reflected in the 𝑥-axis. Find, with respect to the standard basis, the matrix of this combined transformation.

  • A⎡⎢⎢⎢⎣−√3212√3212⎤⎥⎥⎥⎦
  • B⎡⎢⎢⎢⎣−12−√32−√3212⎤⎥⎥⎥⎦
  • C⎡⎢⎢⎢⎣12−√32√32−12⎤⎥⎥⎥⎦
  • D⎡⎢⎢⎢⎣12√32√3212⎤⎥⎥⎥⎦
  • E⎡⎢⎢⎢⎣√32−1212−√32⎤⎥⎥⎥⎦

Q14:

Suppose 𝐴 and 𝐵 are 2×2 matrices, with 𝐴 representing a counterclockwise rotation of 30∘ about the origin and 𝐵 representing a reflection in the 𝑥-axis. What does the matrix 𝐵𝐴 represent?

  • Aa reflection in the line through the origin at a 75∘ inclination
  • Ba reflection in the line through the origin at a −15∘ inclination
  • Ca reflection in the line through the origin at a −75∘ inclination
  • Da reflection in the line through the origin at a −45∘ inclination
  • Ea reflection in the line through the origin at a 15∘ inclination

Q15:

Suppose 𝐴 and 𝐵 are 2×2 matrices, with 𝐴 representing a counterclockwise rotation of 30∘ about the origin and 𝐵 representing a reflection in the 𝑥-axis. What does the matrix 𝐴𝐵 represent?

  • Aa reflection in the line through the origin at a −15∘ inclination
  • Ba reflection in the line through the origin at a −75∘ inclination
  • Ca reflection in the line through the origin at a 15∘ inclination
  • Da reflection in the line through the origin at a 75∘ inclination
  • Ea reflection in the line through the origin at a −45∘ inclination

Q16:

Describe the geometric effect of the transformation produced by the matrix 0−330.

  • Aa dilation with center the origin and scale factor 3 followed by a reflection in the line 𝑦=−𝑥
  • Ba dilation with center the origin and scale factor 3 followed by a 180∘ rotation about the origin
  • Ca dilation with center the origin and scale factor 3 followed by a reflection in the line 𝑦=𝑥
  • Da dilation with center the origin and scale factor 3 followed by a 90∘ rotation about the origin
  • Ea dilation with center the origin and scale factor 3 followed by a −90∘ rotation about the origin

Q17:

Which of the following compositions of transformations is represented by the matrix 0−2−20?

  • Aa dilation with center the origin and scale factor 2 followed by a reflection in the line 𝑦=−𝑥
  • Ba rotation by 180∘ about the origin followed by a reflection in the line 𝑦=𝑥
  • Ca dilation with center the origin and scale factor 2 followed by a reflection in the line 𝑦=𝑥
  • Da dilation with center the origin and scale factor −2 followed by a reflection in the line 𝑦=−𝑥
  • Ea dilation with center the origin and scale factor −2 followed by a reflection in the line 𝑦=𝑥

Q18:

A dilation with center the origin is composed with a rotation about the origin to form a new linear transformation. The transformation formed sends the vector 34 to −3356.

Find the matrix representation of the transformation formed.

  • A512125
  • B5−12125
  • C−110014
  • D5−12125
  • E−12.921.441.4412.92

Find the scale factor of the original dilation.

  • Ascale factor = 169
  • Bscale factor = 13
  • Cscale factor = 154
  • Dscale factor = 13
  • Escale factor =√119

Q19:

The unit square, with vertices 𝑂(0,0),𝐴(1,0),𝐵(1,1), and 𝐶(0,1), is transformed by a rotation and then a dilation. Its image under this combined transformation is 𝑂𝐴𝐵𝐶, as shown in the diagram.

What are the coordinates of 𝐴′?

  • A23,23
  • B32,32
  • C32,−32
  • D23,−23
  • E32,32

What is the matrix of the combined transformation?

  • A⎡⎢⎢⎣32−32−3233⎤⎥⎥⎦
  • B⎡⎢⎢⎣3232−3232⎤⎥⎥⎦
  • C⎡⎢⎢⎣23−232323⎤⎥⎥⎦
  • D⎡⎢⎢⎣32−323232⎤⎥⎥⎦
  • E⎡⎢⎢⎣23−23−2323⎤⎥⎥⎦

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